How do I figure this out?

Question

Each boat holds max of 8 people. Each boat can carry a total of 1200 pounds. Assume adults weigh 150, kids 75 lbs. Assuming each boat holds 200lbs. of gear plus 10 lbs. of gear per person. Write an inequality that illustrates the weight limit for a group of adults and children on the boat and a second representing the total # of passengers. Graph the solution set to the inequalities

Answer 1

For this problem you need to track numbers and weights for adults and children, so we need two inequalities and two variables.

 

Let x be the number of adults, and y be the number of children.

 

Each boat holds a maximum of 8 people, so

0 ≤ x + y ≤ 8

 

Each boat can hold a maximum of 1200 pounds, and adults weight 150 lbs, children 75 lbs. x adults will weight 150x lbs, and y children will weigh 75y lbs. The total is then limited by:

0 ≤ 150x + 75y ≤ 1200

 

All numbers must be positive, so you only need to graph the first quadrant.

 

To graph x + y ≤ 8, we look at x + y = 8, and shade below it since the inequality is less than. The easiest way to graph these is to find the x- and y-intercepts. The x-intercept is when y=0, and the y-intercept is when x=0.

 

x-int:

x+0=8

x=8

(8,0)

 

y-int:

0+y=8

y=8

(0,8)

 

Plot the points (0,8) and (8,0) and connect them to make this line.

 

We can do the same for 150x + 75y ≤ 1200

 

x-int:

150x + 75(0) = 1200

150x = 1200

x = 8

(8,0)

 

y-int:

150(0) + 75y = 1200

75y = 1200

y = 16

(0,16)

 

Plot the points (8,0) and (0,16) to make this line.

 

The overall solution to this problem is the part of the graph that is below both lines. The part of the graph between the lines represents solutions where there are more than 8 people, but are still under the weight limit because there are multiple children, and isn't part of the solution.